Hi,
You are no bother whatsoever. That is a good question. First, I don't recommend disconnecting the battery with the engine running. That can cause different issues.
Now, the easiest way to determine if the alternator is good is by using a multimeter or voltmeter. Before starting the engine, check the voltage at the battery. With the engine off, it should be slightly above 12v.
Next, start the engine. With the engine running, the voltage should be closer to 14v if the alternator is working.
Here is a link that explains how to do this in more detail:
https://www.2carpros.com/articles/how-to-check-a-car-alternator
Let me know if that is something you can do or have done. Also, if the vehicle is drivable, most parts stores will test the alternator and battery free of charge.
Let me know.
Take care,
Joe
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Saturday, June 15th, 2024 AT 6:27 PM